The correct answer is option 'D' - Rayleigh number. The Rayleigh number is a non-dimensional number used to determine the transition from laminar to turbulent flow in free convection.

Explanation:
1. Introduction to free convection:
- Free convection is a mode of heat transfer that occurs due to the natural circulation of fluid caused by density differences.
- It takes place when there is no forced flow or external means of fluid movement.
- Free convection commonly occurs when a fluid is heated from below, and the heated fluid expands, becomes less dense, and rises while the cooler fluid sinks.

2. Transition from laminar to turbulent flow:
- In free convection, the flow can be either laminar or turbulent, depending on the conditions.
- Laminar flow is characterized by smooth, orderly motion, while turbulent flow is characterized by chaotic, random motion.
- The transition from laminar to turbulent flow in free convection is determined by a dimensionless number called the Rayleigh number.

3. Rayleigh number (Ra):
- The Rayleigh number is defined as the ratio of the buoyancy force to the viscous force in a fluid flow.
- It is given by the formula: Ra = (gβΔT L^3)/(να), where g is the acceleration due to gravity, β is the volumetric thermal expansion coefficient, ΔT is the temperature difference, L is a characteristic length, ν is the kinematic viscosity, and α is the thermal diffusivity.
- The Rayleigh number represents the relative importance of buoyancy forces compared to viscous forces in the flow.
- For low Rayleigh numbers, the flow is laminar, while for high Rayleigh numbers, the flow becomes turbulent.

4. Importance in free convection:
- The Rayleigh number is particularly important in free convection because it determines the flow regime and heat transfer characteristics.
- When the Rayleigh number is below a critical value, the flow is laminar, and heat transfer occurs primarily through conduction.
- As the Rayleigh number exceeds the critical value, the flow becomes turbulent, leading to enhanced heat transfer through convective mixing and increased turbulence.
- Therefore, the Rayleigh number is used as a criterion to predict the transition from laminar to turbulent flow in free convection.

In conclusion, the Rayleigh number is the non-dimensional number used to determine the transition from laminar to turbulent flow in free convection. It represents the relative importance of buoyancy forces to viscous forces in the flow and plays a crucial role in predicting the flow regime and heat transfer characteristics.